Convexity theory

See Blackwell and Girshiek, Theory of Games and Statistical Decisions, Chap. 2, John Wiley A Sons, Inc., New York, 1954, for a more complete discussion of convexity. 

Let II II denote the Euclidean norm and define = gk+i gk- Table I provides a chronological list of some choices for the CG update parameter. If the objective function is a strongly convex quadratic, then in theory, with an exact line search, all seven choices for the update parameter in Table I are equivalent. For a nonquadratic objective functional J (the ordinary situation in optimal control calculations), each choice for the update parameter leads to a different performance. A detailed discussion of the various CG methods is beyond the scope of this chapter. The reader is referred to Ref. [194] for a survey of CG methods. Here we only mention briefly that despite the strong convergence theory that has been developed for the Fletcher-Reeves, [195],... 

Unlike solid electrodes, the shape of the ITIES can be varied by application of an external pressure to the pipette. The shape of the meniscus formed at the pipette tip was studied in situ by video microscopy under controlled pressure [19]. When a negative pressure was applied, the ITIES shape was concave. As expected from the theory [25a], the diffusion current to a recessed ITIES was lower than in absence of negative external pressure. When a positive pressure was applied to the pipette, the solution meniscus became convex, and the diffusion current increased. The diffusion-limiting current increased with increasing height of the spherical segment (up to the complete sphere), as the theory predicts [25b]. Importantly, with no external pressure applied to the pipette, the micro-ITIES was found to be essentially flat. This observation was corroborated by numerous experiments performed with different concentrations of dissolved species and different pipette radii [19]. The measured diffusion current to such an interface agrees quantitatively with Eq. (6) if the outer pipette wall is silanized (see next section). The effective radius of a pipette can be calculated from Eq. (6) and compared to the value found microscopically [19]. 

Figure 15 Comparison of theory and experiment for the fractionation of oligoade-nylates on ion exchange materials, (a) Simulated chromatogram, (b) Observed chromatogram. An example of how theory is being used to attempt to optimize performance of ion exchange materials. The curve in (a) shows the nonlinear gradient development with a convex curvature. (Reproduced with permission of Elsevier Science from Baba, Y., Fukuda, M., and Yoza, N., J. Chromatogr., 458, 385, 1988.)...

However, bubble nonhomogeneous distribution exists in two-phase shear flow. As yet, the following general trends in void fraction radial profiles are being identified for bubbly upward flow (Zun, 1990) concave profiles (Serizawa et al., 1975) convex profiles (Sekoguchi et al., 1981), and intermediate profiles (Sekoguchi et al., 1981 Zun, 1988). Two theories are currently dominant ... 

The concept of convexity is useful both in the theory and applications of optimization. We first define a convex set, then a convex function, and lastly look at the role played by convexity in optimization. 

For example, it is usually impossible to prove that a given algorithm will find the global minimum of a nonlinear programming problem unless the problem is convex. For nonconvex problems, however, many such algorithms find at least a local minimum. Convexity thus plays a role much like that of linearity in the study of dynamic systems. For example, many results derived from linear theory are used in the design of nonlinear control systems. 

Molecular Theory of Surface Tension (Harasima) Molecules, Barriers to Internal Rotation in (Wilson) Molecules, Convex, in Gaseous and Crystalline States (Kihara). ... 

Many methods in chemistry for the correlation energy are based on a form of perturbation theory, but the positivity conditions are quite different. Traditional perturbation theory performs accurately for all kinds of two-particle reduced Hamiltonians, which are close enough to a mean-field (Hartree-Fock) reference. There are a myriad of chemical systems, however, where the correlated wave-function (or 2-RDM) is not sufficiently close to a statistical mean field. Different from perturbation theory, the positivity conditions function by increasing the number of extreme two-particle Hamiltonians in which are employed as constraints upon the 2-RDM in Eq. (50) and, hence, they exactly treat a certain convex set of reduced Hamiltonians to all orders of perturbation theory. For the... 

Y. Nesterov and A. S. Nemirovskii, Interior Point Polynomial Method in Convex Programming Theory and Applications, SIAM, Philadelphia, 1993. 

Imprecise probabilities Any of several theories involving models of uncertainty that do not assume a unique underlying probability distribution, but instead correspond to a set of probability distributions. An imprecise probability arises when one s lower probability for an event is strictly smaller than one s upper probability for the same event. Theories of imprecise probabilities are often expressed in terms of a lower probability measure giving the lower probability for every possible event from some universal set, or in terms of closed convex sets of probability distributions (which are generally much more complicated structures than either probability boxes or Dempster-Shafer structures). 

A qualitative description of Eyring et al s approach to the "curved-front theory is given by Taylor (Ref 3, pp 150-52) in order to avoid the algebraic complexity of the theory. It has been argued that the effect of lateral pressure-losses should cause the wave front to become curved into a lens-shaped figure convex at the front and not remain plane as... 

To give perspective to /( S ) when S is defined for the continuous p(x) considered before, we mention the following facts, provable by the elementary principles of convexity, calculus of variations, and moment theory. 

Naturally, the full proof of the above statements concerning M, the information-minimizing F, and the limit properties, which take us into the general theory of convexity, the Hahn-Banach theorem, and delicate minimization methods, are beyond the scope of this paper, and are being dealt with elsewhere. But once their theoretical justification is established, it is quite striking to see how simple is the formal derivation of an explicit expression of the first right-hand term in Eq. (3). This will be seen later on. 

We will consider the cold-gas-convex surface of the flame front as a curved cell of the flame which had been formed after the plane flame lost its stability. The steady state of the convex flame is a result of the nonlinear hydrodynamic interaction with the gas flow field (see Zeldovich, 1966, 1979). In the linear approximation the flame perturbation amplitude grows in time in accordance with Landau theory, but this growth is restricted by nonlinear effects. 

Figure 5. Examples of moment free energy (70) for Flory-Huggins theory of length-polydisperse polymers, with one moment density, p, retained. The parent is of the Schulz form (65), with pf = 0.03, Lu = 100 (hence p = p, /Lv = 3 x 10-4), and a = 2 (hence Lw = 150) the point pt = pj° is marked by the filled circles. In plot (a), the value of x = 0.55 is sufficiently small for the parent to be stable The moment free energy is convex. Plot (b) shows the cloud point, % 0.585, where the parent lies on one endpoint of a double tangent the other endpoint gives the polymer volume fraction p, in the shadow phase. Increasing x further, the parent eventually becomes spinodally unstable [x 0.62, plot (c)]. Note that for better visualization, linear terms have been added to all free energies to make the tangent at the parent coincide with the horizontal axis.

Part 1, comprised of three chapters, focuses on the fundamentals of convex analysis and nonlinear optimization. Chapter 2 discusses the key elements of convex analysis (i.e., convex sets, convex and concave functions, and generalizations of convex and concave functions), which are very important in the study of nonlinear optimization problems. Chapter 3 presents the first and second order optimality conditions for unconstrained and constrained nonlinear optimization. Chapter 4 introduces the basics of duality theory (i.e., the primal problem, the perturbation function, and the dual problem) and presents the weak and strong duality theorem along with the duality gap. Part 1 outlines the basic notions of nonlinear optimization and prepares the reader for Part 2. 

Several theories relating to emulsion type have been proposed. The most satisfactory general theory of emulsion type is that originally proposed for emulsions stabilised by finely divided solids (see Figure 10.1). If the solid is preferentially wetted by one of the phases, then more particles can be accommodated at the interface if the interface is convex towards that phase (i.e. if the preferentially... 

It is generally not possible to perceive a convex surface profile, given by the free vortex theory. For most practical purposes the water surface may be supposed to be a straight line from A to B, raised at the outside wall and depressed at the inside, with the slope given by the ordinary superelevation formula used for highway curves, tan 0 = Vz/gr, where r is the radius of the curve to the center of the channel. 

CHALLENGING MATHEMATICAL PROBLEMS WITH ELEMENTARY SOLUTIONS, A.M. Yaglom and I.M. Yaglom. Over 170 challenging problems on probability theory, combinatorial analysis, points and lines, topology, convex polygons, many other topics. Solutions. Total of 445pp. 5S 8b. Two-vol. set. 

Relation between surface tension and the pressure differences across a curved liquid surface. We must now return to a most important consequence of the existence of free surface energy, which was known to Young and Laplace, and is the foundation of the classical theory of Capillarity, and of most of the methods of measuring surface tension. If a liquid surface be curved the pressure is greater on the concave side than on the convex, by an amount which depends on the surface tension and on the curvature. This is because the displacement of a curved surface, parallel to itself, results in an increase in area as the surface moves towards the convex side, and work has to be done to increase the area. This work is supplied by the pressure difference moving the surface. 

At one time a very attractive theory of the inversion of emulsions, called the oriented-wedge theory, was rather generally accepted. Starting from the fact that the soaps of divalent metals usually form water-in-oil emulsions, while those of monovalent metals form oil-in-water emulsions, it was suggested7 that, since in the divalent soaps there are two hydrocarbon chains attached to one metal atom, while in the monovalent there is only one chain, the molecules of these soaps are wedge-shaped, being wider at the water-soluble end in the case of the monovalent soaps, and at the oil-soluble end with the di- and trivalent soaps. A closely packed layer of these molecules would therefore naturally curve with the concave side towards the oil in the case of the monovalent soaps, with the convex side towards the oil in the case of the divalent soaps thus the... 

The mixing rules for the hard convex body part of the GvdW-EOS are given by theory [7]. The most significant parameter for the attractive part of the GvdW-EOS is the critical temperature. For the mixture, this is calculated for an equivalent substance as in the previous paper [2] ... 

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